Arithmetic persistence conjecture

Let $k\subset L$ be an extension of algebraically closed fields of characteristic zero, and let $X$ be an arithmetically hyperbolic finite type separated scheme over $k$.

Arithmetic Persistence Conjecture. The base change $X_L$ is arithmetically hyperbolic over $L$.

Under the Lang–Vojta arithmetic conjecture, this persistence statement follows for projective varieties from the corresponding persistence of grouplessness. The source states it more generally for finite type separated schemes and gives no proof in that generality.